Question

Give an example of a function that does not have an inverse function.

Short answer

An example of a function that does not have an inverse is the squaring function \( f(x) = x^2 \) under real numbers, because it is not one-to-one.

Step-by-step solution

Identify a Non-Injective Function

A non-injective function (or not one-to-one) includes repetitions in the output for different inputs, meaning that two or more distinct elements in the domain correspond to the same element in the codomain. A common example is the squaring function \( f(x) = x^2 \) defined for real numbers. For instance, both -2 and 2 give the same output 4.

Explain Why the Function Is Not Invertible

An inverse function, if existed, would map back the element 4 to both -2 and 2, which contradicts the definition of a function. Therefore, the squaring function \( f(x) = x^2 \) does not possess an inverse function under the set of real numbers.